coupled with fast matrix–vector multiply and compression algorithms, and iterative solvers such as gmres, provide spectacularly effective computational tools for solving a range of linear boundary value problems, for example in potential theory, elasticity, and acoustic and electromagnetic wave scattering ...
Therefore, many efforts have been made to obtain partial results or to develop algorithms to get the answer of this question (for instance, see [11]). Before continuing, let us recall some definitions and notations used in numerical semigroups. Let Z and N be the set of integers and the...
I actually made a brand new Graph Theory API for it, to fit my needs. But I use another one I did between 2006 and 2009, to manage what is called application resources. This was a very interesting management library called Veda. In the case of Math Touch Book, I just re-used it to...
” 0 2 —Richard Feynman [1, p.161] 0 / h p 1. Introduction - t n A New Kind of Science [2], the 1280treatise by Mathematica creator Stephen Wol- a fram, has only a few things to say about quantum computing. Yet the book’s goal—to u q understand nature in computational ...
My archives record that in late December I was taking a more computation-first approach, and considering the title Algorithms in Nature: An Introduction to Complexity. But soon I was submerged in the intense effort to develop Mathematica 2.0, and this is what consumed me for most of 1990—...
In Section 2, we describe the characteristics of an OKP system and why CPN-CS is needed for modeling and control of the system. CPN-CS is defined in Section 3. Algorithms, by which the structure of a CPN may be updated, are provided in Section 4. In Section 5, we present a case ...
Of NKS-based models now in use, the vast majority are still based on cellular automata—the very first simple programs I studied back in the early 1980s. There’s been some use of substitution systems, mobile automata and various graph-based systems. But cellular automata are still definite...
I could tell as soon as I did this that there were interesting things going on, but it took a couple more years before I began to really appreciate the force of what I’d seen. For me it all started with one picture: Or, in modern form: ...
In this case, the problem’s complexity becomes exponential, that’s because the number of sets that we need to sum and check will grow quickly even with smaller n. Every P problem is also an NP problem, so the class P is a subclass of class NP. NP-complete Problems: These are the...
In this case, the number of the solutions of the congruence equation is 2(p−2). So in the case (B), the number of all solutions of the congruence equation (9) is 3 × 2(p−2) = 6(p−2). It is clear that the number of the solutions of the congruence equation d + e...